Theorem of Pythagoras
The theorem of Pythagoras for the relation between the length of the sides of a right triangle is well known by any high school student. Although the theorem is simple it is one of the very important theorems in mathematics, because the definition of distance in an Euclidean space is based upon this theorem. Any one using this theorem should therefore once have seen a proof. There are numerous proofs, but one very nice geometrical proof is shown here.
 Theorem of Pythagoras

Given a rightangled triangle with sides a,b,c and c the hypotenuse (the side opposite the right angle) then
\[
c^2 = a^2 + b^2
\]
 Proof

Assume that length and angle are invariant under translation and rotation. Let a rightangled triangle with sides a,b,c be translated and rotated in the following four positions to form a square of side c:
The total area of the square $c^2$ is equal to the shaded area, $(ba)^2$, plus the area of the four triangles.
If the rightangled triangles are translated and rotated to form the rectangle, we see that its area is $2ab$.
Now we have:
\begin{align*}
c^2&=(ba)^2+2ab \\
&=b^2+a^22ab+2ab \\
&=a^2+b^2
\end{align*}
An alternative approach is shown below. Before the area of the two right triangles on the left and the shaded square is $c^2+2ab$, after the swap of these two triangles into the shaded square the total area remains the same but is now equal to $a^2+b^2+2ab$ from which follows $c^=a^2+b^2$.
Source:Wikipedia 
Q.E.D.
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